12/5/2004(c) 2001-4, University of Washington21-1 - PDF document

CSE143 Au0421-1 12/5/2004(c) 2001-4, University of Washington21-1 Binary Search Trees 12/5/2004(c) 2001-4, University of Washington21-2 Costliness of contains Idea: order the nodes in the tree so that, given that a All nodes in its left subtreecontain values All nodes in its right subtreec&#x 000;ontain values A binary tree with these properties is called a binary (BST) CSE143 Au0421-3 12/5/2004(c) 2001-4, University of Washington21-9 Cost of BST Work done at each node:Number of nodes visited (depth of recursion): 12/5/2004(c) 2001-4, University of Washington21-10 Must preserve BST invariant: insert new element in correct place in BSTTwo base casesTree is empty: create new node which becomes the root of the If node contains the value, found it; suppress duplicate addRecursive caseCompare value to current node’s valueIf value t node's value, add to left subtreerecursivelyOtherwise, add to right subtreerecursively 12/5/2004(c) 2001-4, University of Washington21-11 Add 8, 10, 5, 1, 7, 11 to an initially empty BST, in that 12/5/2004(c) 2001-4, University of Washington21-12 What if we change the order in which the numbers are Add 1, 5, 7, 8, 10, 11 to a BST, in that order (following the algorithm): CSE143 Au0421-5 12/5/2004(c) 2001-4, University of Washington21-17 Algorithm: find the node contairemoved, and remove that node from the treeRemoving a leaf node is easy: replace with an empty treeRemoving a node with only one non-empty subtreeis easy: How to remove a node that has two non-empty subtrees?Need to pick a new element to be the new root node, and adjust at least one of the subtreesE.g., remove the largest element of the left subtree(will be one of the easy cases described above), make that the new root 12/5/2004(c) 2001-4, University of Washington21-18 Cost of operations is proportional to height of treeBest case: tree is Depth of all leaf nodes is roughly the sameIf tree is unbalanced, height can be as bad as the number of nodes in the treeTree becomes just a linear list 12/5/2004(c) 2001-4, University of Washington21-19 A binary search tree is a good general implementation of a set, if the elements can be orderedBoth contains and add benefit from divide-and-conquer Good properties depend on the tree being roughly balancedNot covered (or, why take a data structures course?)How are other operations implemented (e.g. iterator, remove)?How do you keep the tree balanced as items are added and